The CP-matrix Approximation Problem

TL;DR

This paper addresses the problem of approximating matrices with completely positive matrices by formulating it as a linear optimization problem and proposing a semidefinite algorithm for its solution.

Contribution

It introduces a new formulation for the CP-matrix approximation problem using linear constraints and cone of moments, along with a semidefinite algorithm for solving it.

Findings

The proposed algorithm effectively projects matrices onto the CP cone.

A CP-decomposition can be obtained when the projection problem is feasible.

The method extends existing techniques for CP-matrix approximation.

Abstract

A symmetric matrix $A$ is completely positive (CP) if there exists an entrywise nonnegative matrix $V$ such that $A = V V ^T$. In this paper, we study the CP-matrix approximation problem of projecting a matrix onto the intersection of a set of linear constraints and the cone of CP matrices. We formulate the problem as the linear optimization with the norm cone and the cone of moments. A semidefinite algorithm is presented for the problem. A CP-decomposition of the projection matrix can also be obtained if the problem is feasible.